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PHD, subpixel guiding, and star SNR

Q: I've seen some quote the accuracy of the guiding available in their software with numbers like 1/20th of a pixel accuracy. a) How is this possible, and b) How accurate is PHD?

There are a whole bunch of things that will go into how accurate your guiding is. Your mount, the wind, flex, etc. all go into this. Here, we'll go over finding the star's position to an accuracy well below a pixel and how the star's signal to noise ratio (SNR) affects this accuracy. Since the SNR of the star (which is determined by the amount of light hitting the CCD and by the camera's noise) affects the accuracy, I won't quote a single hard and fast number as to how accurate it can be. I could quote an extreme (1/200th of a pixel) but I doubt it would mean much and would sound more like marketing hype. Since PHD is free, there tends to be little incentive for marketing hype.

In an article I wrote for Astronomy Technology Today on PHD Guiding, I went over the basics of how you find the middle of a star to accuracy well below a pixel. Here's an image that may help. Below we have stars hitting three different 2x2 areas of a CCD. In the first case, the star is hitting the exact intersection of these four pixels, so the star's light is evenly distributed among all four. In the next, we have the star still centered vertically but shifted to the right just a bit. More of the star's energy is now on the right two pixels than the left, so these two are brighter than the left two. For both the left and the right, the top and bottom pixels have gotten the same amount of energy, so there is no top/bottom difference. In the last panel, the star is now centered a bit down and to the right of the intersection of the four pixels. Most of its energy hits the lower-right pixel with equal amounts hitting the lower-left and upper-right and the least hitting the upper-left. Thus, so long as the star lights up more than one pixel, we can estimate small shifts in its position.

Here's another way to look at this. In this surface plot, I'm showing the energy of a simulated star that is perfectly centered.

Here is that same star shifted just a touch -- 0.2 pixels one way and 0.7 pixels in the other:

Note now now the star's profile is no longer symmetric. We can use this shift to estimate just how much the star has moved. When using perfect images such as these, it's easy to pick up small shifts. The real question is how well it does on real-world stars. I can't exactly go and take pictures of stars that have moved 0.001 pixels or anything of the sort because I'd have no way of knowing how much the star really moved. (I don't even know if the Hubble could hit that...). What I can do is to simulate real stars with real noise and see how well it works.

Here are profiles of the same star from two different cameras (there are actually two stars and one camera is rotated 180 degrees relative to the other so the second star you see clearly to the right on the image on the right is to the left in the noisy one on the left). Clearly, the one on the left has more noise and a lower SNR than the one on the right. These are real profiles from real stars with real cameras (mag 11.7 star, 2 s exposure, Celestron CPC 1100 XLT at f/6.3-ish).

Here, for reference, is an image of the star on the left. This is certainly not a very clean star to guide on, but PHD will lock onto it and guide off of it.

To simulate this, I can get something pretty close by modeling a star with variable amounts of Gaussian noise:

I apologize for the fact that the size of the X and Y axes differ between the simulations and the actual stars. The simulations show a more "zoomed out" view than the actual star profiles, but the SNRs of the two setups are comparable. The big question is how accurately can PHD locate the center of the star in each kind of image?

To get at this, I moved the simulated star by 0.05 pixel steps in both directions so I would know the true distance the star moved (using good old Pythagoras) and I would have a bunch of samples to get a good estimate of PHD's error with no noise, low noise (right image), and high noise (left image). Shown here is the average error in location for the three noise conditions (error bars show standard error of the mean)

Without any noise, PHD is accurate down to (on average) 0.004 pixels or 1/250th of a pixel. With a low amount of noise, the accuracy goes to 0.018 pixels or 1/56th of a pixel and with high amounts of noise it goes to 0.18 pixels or 1/5.5th of a pixel. Better stars and/or better guide cameras will get you more accuracy, but even with this very noisy star, we're still at 1/5th of a pixel accuracy.

What about noise reduction? Does smoothing the image help? Well, it helps in terms of increasing the odds that PHD will lock onto the star, but it doesn't help the accuracy of localization at all. At best, it does nothing and at worst, it hurts your accuracy a bit. Again, it will help PHD find the star in the first place, so if this plot included "lost stars" as errors (and they are errors), it would have a nice effect. But once found, smoothing does nothing to help the localization.

So there you have it. Can we get to sub-pixel levels? Sure. Under perfect conditions, PHD Guiding gets you down to an insane level of 0.004 pixels worth of error showing that the basic math in sub-pixel error calculation works. Done well (and yes, it can be done poorly - I know from first-hand experience a few ways to do it poorly and/or to introduce systematic biases in the star-finding routine), it can get you more accurate star localization than you'd ever really need. The SNR of your guide star will affect your accuracy, however. Under more real-world conditions, accuracy drops, but still can be very good. I have seen many stars in my guide frame with SNRs like the Low-noise SNR star used here that led to a 1/50th pixel accuracy. One notable implication of this is that with higher SNR stars, you can afford to use wider FOV guide setups since the higher SNR leads to increased precision in localization.

In practice, the High Noise star here is about as noisy a star as you can successfully guide on with PHD. Much worse than this and you're going to be suffering lost locks occasionally. With even this level of noise, we're still below a quarter of a pixel worth of error. Odds are, your mount, the wind, flex, etc. will be causing more trouble than even this error unless you're using a very wide FOV guide setup.